Number-Grid Puzzle: Multiply & Add

[Leroy Quet]

I posted the below to sci.math yesterday. Others who replied found many 
solutions for n =3.

I am wondering what the sequence is, where the n_th term is the number of 
solutions for the n-by-n grid.

Thank you,
Leroy Quet

-- 

>           For a positive integer n, put the integers 1 through n^2, one 
>integer
>           per cell, into an n-by-n grid, such that:
>
>           the sum of the products of the elements in each row =
>           the sum of the products of the elements in each column.
>
>           Here is an example, which is NOT a solution, but hopefully will
>           clear-up any confusion as to what I am asking:
>
>           n = 3:
>
>           1  9  3
>
>           6  4  7
>
>           5  8  2
>
>           products of rows: 27, 168, 80,
>           which when summed is 275;
>
>           products of columns: 30, 288, 42,
>           which when summed is 360.
>
>           Since 275 does not equal 360, this is not a solution.
>
>           For which n are there solutions?
>
>           I don't know, myself, if any n >= 2 have a solution, since I 
>haven't
>           tried too hard yet to find any solutions.
>
>           Thanks,
>           Leroy Quet